Question

How do you graph the polar equation `r=sin4theta`?

Answer 1

See the graph and the explanation.

Explanation:

If p an q are co-prime positive integers, the number of petals

created by either `r = sin (p/q)theta or r = cos (p/q)theta` is p.

Here. `r = sin 4theta` and p = 4. Use

`r = sqrt( x^2 + y^2 ), ( x, y ) = r ( cos theta, sin theta )`.and

`sin 4theta = 4C_1cos^3 theta sin theta - 4C_3 cos theta`

`= 4sin theta cos theta ( cos^2theta - sin^2theta )`

and obtain the Cartesian form of `r = sin 4theta` as

`( x^2 + y^2 )^2.5 - 4 x y ( x^2 - y^2 ) = 0`.

The Socratic graph is immediate.

graph{ ( x^2 + y^2 )^2.5 - 4 x y ( x^2 - y^2 ) = 0[-2.2 2.2 -1.1 1.1] }

The number of petals of `r= sin (4/3)theta` is also 4#. See the idiosyncratic petals.

graph{3xy(x^2-y^2)-4(x^2+y^2)^2sqrt(1-(x^2+y^2))(1-2(x^2+y^2))=0[-4 4 -2 2]}.